There're different approaches to build ZKP for different statements.
E.g., there'e ad-hoc protocols: Schnorr protocol allows you to build a proof of knowledge of discrete logarithm of some group element.
There're also universal ZKP protocols, which allows you to build a proof for any statement, formulated as a computational circuit. This is a relatively new field of research, and examples of efficient protocols are STARKs, Bulletproofs, ZkSNARKS. You can google for them, but I should warn you in advance that
unlike ad-hoc solutions, all these universal protocols are pretty complex and not-easy to understand.
If your goal is only to proof that you know discrete logarithm of some of
ECC points from the list, you can use this simple ad-hoc solution,
a modification of well-known Schnorr protocol:
(search for OR-proof there).
Also, as I understand, you're looking for solution to prove membership of party in some group, and make it anonymously. For this goal,
there's a crypto-primitive called "group signatures", or "ring signature", and "linkable group/ring signatures". E.g., ring signature allows you to sign a message with your secret key, so that everybody can see that the signature is correct and belong to some member of the group, and at the same time, nobody knows who exactly signed it (so, it's anonymous).
Linkable group signature scheme allows detecting of 2 signatures of the the person (e.g. it could be useful for voting, in order to prevent a person to vote twice).
Depending on your basic signature scheme and using keys, you should look for corresponding signature schemes with additional features. E.g., if you're using elliptic curve crypto, so that your public keys are points on elliptic curve, you could look into Schnorr Ring Signature Scheme.